Extended King integral for modeling of parametric array loudspeakers with axisymmetric profiles

Parametric array loudspeakers have been widely used in audio applications for generating directional audio beams. However, accurately calculating audio sound with a low computational load remains challenging, even for basic axisymmetric source profiles. This work addresses this challenge by extending the King integral in linear acoustics to incorporate both cumulative and local nonlinear effects, under the framework of the quasilinear solution without the paraxial approximation. The proposed method exploits the azimuthal symmetry in cylindrical coordinates to simplify modeling. To further improve computational efficacy, fast Hankel and Fourier transforms are employed for the radial and beam radiation directions, respectively. Numerical results with both uniform and focusing profiles demonstrate the advantages of the proposed approach over the traditional spherical wave expansion and direct integration methods, especially for larger aperture sizes. Specifically, for typical configurations with source aperture size of 0.2 m, we observe at least a 24-fold improvement in computational speed and a 227-fold reduction in memory requirements. These advancements allow us, for the first time, to present the sound field radiated by parametric array loudspeakers with a large aperture size of up to 0.5 m, without paraxial approximations. The implementation codes are available on https://github.com/ShaoZhe-LI/PAL_King.